Identify the parent ring and principal functional group

The compound is a cyclohexane ring containing an $-OH$ group and two methyl groups on the same carbon.

Since alcohol has higher priority, the parent name must be based on cyclohexanol rather than hydroxycyclohexane.

Number the ring

In cyclic alcohols, the carbon bearing the $-OH$ group gets locant $1$ automatically.

So the parent is: $\text{cyclohexan-1-ol}$ which is commonly written as: $\text{1-cyclohexanol}$

Locate the methyl substituents

From the structure, the two methyl groups are attached to carbon $3$ of the ring.

Hence the substituent part is: $3,3\text{-dimethyl}$

Write the complete name

Combining substituent and parent name: $3,3\text{-dimethyl-1-cyclohexanol}$

Check the options

• A: $3,3$-dimethyl-$1$-cyclohexanol ✅ Correct
• B: $1,1$-dimethyl-$3$-hydroxy cyclohexane ❌ Alcohol should be principal suffix, not hydroxy prefix
• C: $3,3$-dimethyl-$1$-hydroxy cyclohexane ❌ Same issue; should use alcohol suffix
• D: $1,1$-dimethyl-$3$-cyclohexanol ❌ Wrong numbering

Final answer

The correct IUPAC name is: $\boxed{3,3\text{-dimethyl-1-cyclohexanol}}$

So, the correct option is A.